VMO 2022 Problem 7

Let $ABC$ be an acute triangle, $B,C$ fixed, $A$ moves on the big arc $BC$ of $(ABC)$ . Let $O$ be the circumcenter of $(ABC)$ $(B,O,C$ are not collinear, $AB \ne AC)$ , $(I)$ is the incircle of triangle $ABC$ . $(I)$ tangents to $BC$ at $D$ . Let $I_a$ be the $A$ -excenter of triangle $ABC$ . $I_aD$ cuts $OI$ at $L$ . Let $E$ lies on $(I)$ such that $DE \parallel AI$ . a) $LE$ cuts $AI$ at $F$ . Prove that $AF=AI$ . b) Let $M$ lies on the circle $(J)$ go through $I_a,B,C$ such that $I_aM \parallel AD$ . $MD$ cuts $(J)$ again at $N$ . Prove that the midpoint $T$ of $MN$ lies on a fixed circle.